Hello. I'm Professor Von Schmohawk and welcome to Why U.
In our lectures, so far we have discussed multiplying positive numbers.
But what happens when we multiply negative numbers?
With the invention of negative numbers the rules of multiplication had to be expanded
to allow the operands to be either positive or negative.
The rules of multiplication were picked so as to keep everything consistent.
For instance, since one is the multiplicative identity
if a number is multiplied by one
we should expect that the number’s value and sign will not change.
Therefore, multiplying a positive number times a positive number
must produce a positive result
and multiplying a positive number times a negative number must produce a negative result.
Because of the commutative property of multiplication
we should be able to swap the operands and get the same result.
So if either operand is negative we must still get a negative result.
But what if both operands are negative?
If two negative numbers are multiplied, should the product be positive or negative?
Let’s try it both ways and see what happens.
Let’s multiply six minus four, times negative one.
We know what the answer should be.
Six minus four is two.
And we already have shown that the product of a negative and positive number
must be negative.
So the answer must be negative two.
But instead, let’s say we use the distributive property
and multiply negative one times each number in the parentheses separately.
We then have negative one times six
plus negative one times negative four.
We know that negative one times six is negative six.
But we don’t know what sign the product should be when we multiply two negative numbers.
Is negative one times negative four negative four or positive four?
Let’s try both possibilities and see which one gives us the correct answer.
If we assume that multiplying two negative numbers results in a negative product
then we end up adding negative four to negative six
which equals negative ten.
But the answer should be negative two so this is not correct.
The other possibility is that multiplying two negative numbers gives a positive result.
In that case, negative one times negative four would be positive four.
We then add positive four to negative six which gives us negative two
the correct answer.
So, we get the correct answer if we make the rule that
the product of two negative numbers is positive.
Now we know what sign the result should be when we multiply two numbers of any sign.
Multiplying two numbers with the same sign always gives a positive result.
And multiplying two numbers with opposite signs always gives a negative result.
Understanding this, can help us simplify multiplication problems
involving multiple numbers of different signs.
Let's say that we have a bunch of positive and negative numbers which are multiplied.
The commutative property of multiplication
says that we can arrange these numbers in any way we like
so let's group pairs of negative numbers together.
Each pair of negative numbers creates the same result as if the pair was positive
so we can change their signs to positive without changing the result of the multiplication.
Since there was an even number of negative numbers
after each pair of negatives was changed to positives
there were no negative numbers left over.
Therefore, the result of the multiplication is positive.
Now, let's see what happens if we have an odd number of negative numbers.
Once again, we group the negative numbers into pairs and change their signs
but one unpaired negative number is left.
So this time, the result of the multiplication is negative.
Here is another interesting trick which can come in handy.
Multiplying any positive number, which we will call A, by negative one
switches its sign to negative.
Likewise, multiplying any negative number by negative one will switch its sign to positive.
Now, let’s say that we have a sum of several numbers of various signs.
If we enclose the sum in parentheses and multiply by negative one
the distributive property says that this is the same
as multiplying each number individually by negative one
which switches the sign of each number.
So multiplying a sum of numbers in parentheses by negative one
switches the sign of each number.
Instead of multiplying by negative one
we could just put a negative sign in front of the parentheses
which means exactly the same thing.
So a negative sign in front of a parentheses
has the same effect as switching the sign of each number summed in the parentheses.
Here’s one more trick using the distributive property.
Let's say we start with A minus B.
If we enclose this in parentheses with a negative sign in front
it switches the sign of each number.
Using the commutative property, we can then swap the positions of the two numbers
and the result is that we now have B minus A.
So placing a negative sign in front of two numbers which are subtracted
swaps the two numbers.
We have seen that with the invention of negative numbers
the rules of multiplication had to be expanded to allow numbers of any sign to be multiplied.
We have shown that these rules were picked in a way that kept things consistent.
Since positive one is the multiplicative identity
it made sense that multiplying a number by a positive number
should not change that number’s sign.
Since we don’t want to break the commutative property
we must also make the rule that if either one of the operands is negative
the result must be negative.
And if we don’t want to break the distributive property
we must make the rule that if both operands are negative, the result must be positive.
So we now have a set of rules which allow us to multiply integers of any sign.
In the next few lectures we will explore the properties of division
and see how this operation forced the world to create a new type of number.